Let $E$ be a vector space over a field $K$, $x\in E$ and a sequence $\{x_n \}_{n \in \mathbb{N}} \subset E$.
Question:
I need to find a counter example of two different linear functionals $$\psi,\varphi\in E' $$ such that $$\lim_{n \to \infty} \psi(x_n) = \psi(x)$$ but $$\lim_{n \to \infty} \varphi(x_n) \neq \varphi(x)$$
Thanks!
Best Answer
Are you sure this is what you need? This seems to inspire trivial (counter?)examples like the following.
Let $E=\mathbb{R}$, $\psi = 0$, $\varphi(z)=z$ for $z\in \mathbb{R}$. Then take $x_n=0$ for all $n$ and $x=1$.
We obtain $\psi(x_n) = \psi(x) =0$ and $\varphi(x_n) = 0$, $\varphi(x)=1$.